Sculpting brush alpha/texture resolution?
Yes, it can. However, that isn't your issue. In order for texture resolution to be a factor, you would need the edges created by the Dyntopo system to be shorter than the pixels of your texture. That way, there is enough detail in your mesh to create 'steps' between pixels. But as I said, that isn't your issue. From some testing I just did, it appears that using the anchored stroke type doesn't generate new detail. I'm not sure if that is intentional or not, but it seems that is the reason the detail isn't changing.